Given a bargraph B, a border cell of B is a cell of B that shares at least one common edge with an outside cell of B. Clearly, the inner site-perimeter of B is the number of border cells of B. A tangent cell of B is a cell of B which is not a border cell of B and shares at least one vertex with an outside cell of B. In this paper, we study the generating function for the number of k-ary words, represented as bargraphs, according to the number of horizontal steps, up steps, border cells and tangent cells. This allows us to express some cases via Chebyshev polynomials of the second kind. Moreover, we find an explicit formula for the number of bargraphs according to the number of horizontal steps, up steps, and tangent cells/inner site-perimeter. We also derive asymptotic estimates for the mean number of tangent cells/inner site-perimeter.
|Journal||Art of Discrete and Applied Mathematics|
|State||Published - 2021|
Bibliographical notePublisher Copyright:
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- Chebyshev polynomials
- Inner site-perimeter
- K-ary words
ASJC Scopus subject areas
- Applied Mathematics
- Discrete Mathematics and Combinatorics
- Computational Theory and Mathematics